Newspace parameters
| Level: | \( N \) | \(=\) | \( 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1800.v (of order \(4\), degree \(2\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(49.0464475849\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(i)\) |
| Coefficient field: | \(\Q(i, \sqrt{6})\) |
|
|
|
| Defining polynomial: |
\( x^{4} + 9 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 600) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{4} + 9 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} ) / 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} ) / 3 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 3\beta_{2} \)
|
| \(\nu^{3}\) | \(=\) |
\( 3\beta_{3} \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1800\mathbb{Z}\right)^\times\).
| \(n\) | \(577\) | \(901\) | \(1001\) | \(1351\) |
| \(\chi(n)\) | \(-\beta_{2}\) | \(1\) | \(1\) | \(1\) |
Embeddings
For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.
For more information on an embedded modular form you can click on its label.
| Label | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 793.1 |
|
0 | 0 | 0 | 0 | 0 | −7.67423 | + | 7.67423i | 0 | 0 | 0 | ||||||||||||||||||||||||||||
| 793.2 | 0 | 0 | 0 | 0 | 0 | −0.325765 | + | 0.325765i | 0 | 0 | 0 | |||||||||||||||||||||||||||||
| 1657.1 | 0 | 0 | 0 | 0 | 0 | −7.67423 | − | 7.67423i | 0 | 0 | 0 | |||||||||||||||||||||||||||||
| 1657.2 | 0 | 0 | 0 | 0 | 0 | −0.325765 | − | 0.325765i | 0 | 0 | 0 | |||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.c | odd | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1800.3.v.i | 4 | |
| 3.b | odd | 2 | 1 | 600.3.u.b | ✓ | 4 | |
| 5.b | even | 2 | 1 | 1800.3.v.p | 4 | ||
| 5.c | odd | 4 | 1 | inner | 1800.3.v.i | 4 | |
| 5.c | odd | 4 | 1 | 1800.3.v.p | 4 | ||
| 12.b | even | 2 | 1 | 1200.3.bg.n | 4 | ||
| 15.d | odd | 2 | 1 | 600.3.u.g | yes | 4 | |
| 15.e | even | 4 | 1 | 600.3.u.b | ✓ | 4 | |
| 15.e | even | 4 | 1 | 600.3.u.g | yes | 4 | |
| 60.h | even | 2 | 1 | 1200.3.bg.c | 4 | ||
| 60.l | odd | 4 | 1 | 1200.3.bg.c | 4 | ||
| 60.l | odd | 4 | 1 | 1200.3.bg.n | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 600.3.u.b | ✓ | 4 | 3.b | odd | 2 | 1 | |
| 600.3.u.b | ✓ | 4 | 15.e | even | 4 | 1 | |
| 600.3.u.g | yes | 4 | 15.d | odd | 2 | 1 | |
| 600.3.u.g | yes | 4 | 15.e | even | 4 | 1 | |
| 1200.3.bg.c | 4 | 60.h | even | 2 | 1 | ||
| 1200.3.bg.c | 4 | 60.l | odd | 4 | 1 | ||
| 1200.3.bg.n | 4 | 12.b | even | 2 | 1 | ||
| 1200.3.bg.n | 4 | 60.l | odd | 4 | 1 | ||
| 1800.3.v.i | 4 | 1.a | even | 1 | 1 | trivial | |
| 1800.3.v.i | 4 | 5.c | odd | 4 | 1 | inner | |
| 1800.3.v.p | 4 | 5.b | even | 2 | 1 | ||
| 1800.3.v.p | 4 | 5.c | odd | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1800, [\chi])\):
|
\( T_{7}^{4} + 16T_{7}^{3} + 128T_{7}^{2} + 80T_{7} + 25 \)
|
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\( T_{11}^{2} - 4T_{11} - 92 \)
|
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\( T_{17}^{4} - 16T_{17}^{3} + 128T_{17}^{2} + 1216T_{17} + 5776 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( T^{4} + 16 T^{3} + \cdots + 25 \)
|
| $11$ |
\( (T^{2} - 4 T - 92)^{2} \)
|
| $13$ |
\( T^{4} + 729 \)
|
| $17$ |
\( T^{4} - 16 T^{3} + \cdots + 5776 \)
|
| $19$ |
\( T^{4} + 1730 T^{2} + 744769 \)
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| $23$ |
\( T^{4} - 48 T^{3} + \cdots + 76176 \)
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| $29$ |
\( T^{4} + 2120 T^{2} + 446224 \)
|
| $31$ |
\( (T^{2} + 62 T + 865)^{2} \)
|
| $37$ |
\( T^{4} + 32 T^{3} + \cdots + 6400 \)
|
| $41$ |
\( (T^{2} + 56 T + 688)^{2} \)
|
| $43$ |
\( T^{4} + 112 T^{3} + \cdots + 2019241 \)
|
| $47$ |
\( T^{4} - 160 T^{3} + \cdots + 8410000 \)
|
| $53$ |
\( T^{4} - 208 T^{3} + \cdots + 28729600 \)
|
| $59$ |
\( (T^{2} + 6724)^{2} \)
|
| $61$ |
\( (T^{2} - 150 T + 5241)^{2} \)
|
| $67$ |
\( T^{4} - 144 T^{3} + \cdots + 2277081 \)
|
| $71$ |
\( (T^{2} - 136 T + 4528)^{2} \)
|
| $73$ |
\( T^{4} + 224 T^{3} + \cdots + 15366400 \)
|
| $79$ |
\( T^{4} + 19272 T^{2} + 91470096 \)
|
| $83$ |
\( T^{4} - 160 T^{3} + \cdots + 6822544 \)
|
| $89$ |
\( T^{4} + 39968 T^{2} + 369254656 \)
|
| $97$ |
\( T^{4} - 320 T^{3} + \cdots + 137288089 \)
|
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